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If Quantum Computers are not Possible Why are Classical Computers Possible? | Combinatorics and more
As most of my readers know, I regard quantum computing as unrealistic. You can read more about it in my Notices AMS paper and its extended version (see also this post) and in the discussion of Puzzle 4 from my recent puzzles paper (see also this post). The amazing progress and huge investment in quantum computing (that I presented and update  routinely in this post) will put my analysis to test in the next few years.
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november 2017 by nhaliday
gn.general topology - Pair of curves joining opposite corners of a square must intersect---proof? - MathOverflow
In his 'Ordinary Differential Equations' (sec. 1.2) V.I. Arnold says "... every pair of curves in the square joining different pairs of opposite corners must intersect".

This is obvious geometrically but I was wondering how one could go about proving this rigorously. I have thought of a proof using Brouwer's Fixed Point Theorem which I describe below. I would greatly appreciate the group's comments on whether this proof is right and if a simpler proof is possible.

...

Since the full Jordan curve theorem is quite subtle, it might be worth pointing out that theorem in question reduces to the Jordan curve theorem for polygons, which is easier.

Suppose on the contrary that the curves A,BA,B joining opposite corners do not meet. Since A,BA,B are closed sets, their minimum distance apart is some ε>0ε>0. By compactness, each of A,BA,B can be partitioned into finitely many arcs, each of which lies in a disk of diameter <ε/3<ε/3. Then, by a homotopy inside each disk we can replace A,BA,B by polygonal paths A′,B′A′,B′ that join the opposite corners of the square and are still disjoint.

Also, we can replace A′,B′A′,B′ by simple polygonal paths A″,B″A″,B″ by omitting loops. Now we can close A″A″ to a polygon, and B″B″ goes from its "inside" to "outside" without meeting it, contrary to the Jordan curve theorem for polygons.

- John Stillwell
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october 2017 by nhaliday
Correlated Equilibria in Game Theory | Azimuth
Given this, it’s not surprising that Nash equilibria can be hard to find. Last September a paper came out making this precise, in a strong way:

• Yakov Babichenko and Aviad Rubinstein, Communication complexity of approximate Nash equilibria.

The authors show there’s no guaranteed method for players to find even an approximate Nash equilibrium unless they tell each other almost everything about their preferences. This makes finding the Nash equilibrium prohibitively difficult to find when there are lots of players… in general. There are particular games where it’s not difficult, and that makes these games important: for example, if you’re trying to run a government well. (A laughable notion these days, but still one can hope.)

Klarreich’s article in Quanta gives a nice readable account of this work and also a more practical alternative to the concept of Nash equilibrium. It’s called a ‘correlated equilibrium’, and it was invented by the mathematician Robert Aumann in 1974. You can see an attempt to define it here:
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july 2017 by nhaliday
Peter Norvig, the meaning of polynomials, debugging as psychotherapy | Quomodocumque
He briefly showed a demo where, given values of a polynomial, a machine can put together a few lines of code that successfully computes the polynomial. But the code looks weird to a human eye. To compute some quadratic, it nests for-loops and adds things up in a funny way that ends up giving the right output. So has it really ”learned” the polynomial? I think in computer science, you typically feel you’ve learned a function if you can accurately predict its value on a given input. For an algebraist like me, a function determines but isn’t determined by the values it takes; to me, there’s something about that quadratic polynomial the machine has failed to grasp. I don’t think there’s a right or wrong answer here, just a cultural difference to be aware of. Relevant: Norvig’s description of “the two cultures” at the end of this long post on natural language processing (which is interesting all the way through!)
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march 2017 by nhaliday

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